Lattice-Boltzmann type relaxation systems and high order relaxation schemes for the incompressible Navier-Stokes equations

MATHEMATICS OF COMPUTATIONVolume 77, Number 262, April 2008, Pages 943–965S 0025-5718(07)02034-0Article electronically published on December 17, 2007

LATTICE-BOLTZMANN TYPE RELAXATION SYSTEMSAND HIGH ORDER RELAXATION SCHEMES

FOR THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

MAPUNDI BANDA, AXEL KLAR, LORENZO PARESCHI, AND MOHAMMED SEAID

Abstract. A relaxation system based on a Lattice-Boltzmann type discretevelocity model is considered in the low Mach number limit. A third order re-laxation scheme is developed working uniformly for all ranges of the mean freepath and Mach number. In the incompressible Navier-Stokes limit the schemereduces to an explicit high order finite difference scheme for the incompress-ible Navier-Stokes equations based on nonoscillatory upwind discretization.Numerical results and comparisons with other approaches are presented forseveral test cases in one and two space dimensions.

1. Introduction

Many kinetic equations or discrete velocity models of kinetic equations yield,in the limit for small Knudsen and Mach numbers, an approximation of the In-compressible Navier-Stokes (INS) equations. A classical example is given by thediscrete velocity models used for Lattice-Boltzmann methods; see [6, 9, 19, 10, 8].These discrete velocity models can be viewed as relaxation systems for the INSequations.

Relaxation type schemes have been used successfully to discretize such relaxationsystems. In particular, a large number of numerical methods for kinetic equationswith stiff relaxation terms have been considered in fluid dynamics or diffusive lim-its. For these relaxation methods and asymptotic-preserving methods, we refer to[11, 7, 26, 23, 24, 25, 28] and for more general applications of relaxation schemes werefer to the recent review paper [18]. We mention here that, in the context of hy-perbolic conservation laws relaxation schemes are closely related to central schemes[14, 37, 1, 22, 34, 31, 30, 41], in the sense that both approaches provide efficienthigh resolution and Riemann solver free numerical methods for hyperbolic conser-vation laws. Applications of central schemes to problems with stiff sources havebeen considered in [35, 15].

The aim of the present paper is to present a methodology for developing com-putational schemes for INS based on an appropriate discretization of a so-called

Received by the editor November 10, 2005 and, in revised form, January 15, 2007.2000 Mathematics Subject Classification. Primary 76P05, 76D05, 65M06, 35B25.Key words and phrases. Lattice-Boltzmann method, relaxation schemes, low Mach number

limit, incompressible Navier-Stokes equations, high order upwind schemes, Runge-Kutta methods,stiff equations.

This work was supported by DFG grant KL 1105/9-1 and partially by TMR project “Asymp-totic Methods in Kinetic Theory”, Contract Number ERB FMRX CT97 0157.

c©2007 American Mathematical SocietyReverts to public domain 28 years from publication

943

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944 M. BANDA, A. KLAR, L. PARESCHI, AND M. SEAID

relaxation system; see Section 2 for more details of such a system. Such a re-laxation system is derived from a Lattice-Boltzmann type discrete velocity modelwith diffuse scaling. The analytical derivations also demonstrate a relationship be-tween the Lattice-Boltzmann method and such relaxation-based schemes. On theother hand one may refer to [30, 31, 32, 33], and references therein, for alternativeapproaches based on Godunov type methods.

In the present paper a third order relaxation scheme is developed. The schemeworks with uniform accuracy with respect to the Knudsen and Mach numbers,and in the low Mach number limit it reduces to a third order explicit scheme forthe INS equations. This is achieved by combining the ideas developed in [29, 24,25] with third order nonoscillatory spatial discretizations and IMEX Runge-Kuttatime discretizations [4, 38]. The high order nonoscillatory upwind method for theconvective part of the relaxation system turns, in the INS limit (the relaxed scheme),into a high order treatment of the nonlinear convective parts of the INS equations.Clearly, to obtain only a discretization of the limit INS equations, one can use theabove mentioned spatial discretization on the relaxation system and apply any highorder time discretization directly to the resulting semi-discrete relaxed schemes.This allows us to obtain high order INS solvers with better stability properties.

The rest of the paper is organized in the following way. Section 2 contains theLattice-Boltzmann type discrete velocity model and its equivalent associated closedmoment system relaxing to the INS equations. Some simplified relaxation systemsare also presented. In particular we introduce a simplified relaxation system that issuitable to provide relaxed schemes for the incompressible Navier-Stokes equations.Section 3 describes the time and space discretizations and includes a discussion ofthe discretization of the limit equations that originate from the schemes. Finally,Section 4 contains a numerical investigation of the schemes and a comparison withseveral different approaches in one and two space dimensions.

2. Lattice-Boltzmann type discrete velocity models

and simplified relaxation systems

2.1. The Lattice-Boltzmann moment system. The two-dimensional kineticequation

(1)∂f

∂t+ v · ∇f = J(f)

describes the evolution of a particle density f(x,v, t) with x = (x, y) ∈ R2 and

v = (v1, v2) ∈ R2. The left hand side of (1) represents free transport of the

particles, while the right hand side describes interactions through collisions. Fordiscrete models in 2D we have

v ∈ {c0, . . . , cN−1}, ci ∈ R2.

Here we consider a model with nine velocities (N = 9)

c1 = ( 10 ) , c2 = ( 0

1 ) , c3 =(−1

0

), c4 =

(0−1

),

c5 = ( 11 ) , c6 =

(−11

), c7 =

(−1−1

), c8 =

(1−1

),

and c0 = 0. In the discrete case, the v-dependence of the particle distributionf(x,v, t) is uniquely determined through N functions

fi(x, t) = f(x, ci, t), i = 0, . . . , N − 1.


LATTICE-BOLTZMANN TYPE SYSTEMS AND HIGH ORDER SCHEMES 945

Macroscopic quantities like mass–, momentum– or energy–density are obtained bytaking velocity moments of f . If ζ is any v-dependent function, we denote thediscrete velocity integral by

〈ζ〉 =N−1∑i=0

ζ(ci).

Mass and momentum density are then given by

(2) ρ(x, t) = 〈f(x,v, t)〉 and ρu(x, t) = 〈vf(x,v, t)〉 .

In the following we denote the components of the velocity by u = (u1, u2). InLattice-Boltzmann applications, the collision operator J(f) in (1) is typically ofBGK-type

(3) J(f) = −1τ

(f − feq).

The parameter τ > 0 is called relaxation time and feq is the equilibrium distribution.In the isothermal case, feq depends on f through the parameters ρ and u whichare calculated according to (2); see for example [19, 20, 42]. For the standardD2Q9-model [40] with 9 velocities, we have

feq[ρ,u](v) = ρ

(1 + 3u · v − 3

2|u|2 +

92(u · v)2

)f∗(v),

where f∗ is defined by

f∗(ci) =

⎧⎪⎨⎪⎩

49 , i = 0,19 , i = 1, . . . , 4,136 , i = 5, . . . , 8.

The equilibrium distribution is constructed in such a way that

〈J(f)〉 = 0 and 〈vJ(f)〉 = 0,

which reflects conservation of mass and momentum in the collision process.In order to obtain a relation between the kinetic equation (1) and the incom-

pressible Navier-Stokes system, we introduce the diffusive scaling x → x/ε, t → t/ε2

together with a rescaling of velocity u → εu. This scaling describes the small Knud-sen and low Mach number limit of kinetic equations; see [44, 12, 5, 21, 1] for details.Under these transformations, (1) turns into

(4)∂f

∂t+

1εv · ∇f = − 1

ε2τ(f − feq[ρ, εu]).

In our case, (4) consists of nine equations for the occupation numbers f0, . . . , f8.In order to get closer in notation to the Navier-Stokes system, we transform (4)into an equivalent set of moment equations (see also [29, 13] for a similar approach)



using moments based on the following v-polynomials [17]:

P0(v) = 1,

P1(v) =v1

ε, P2(v) =

v2

ε,

P3(v) =v21

ε2− 1

3ε2, P4(v) =

v1v2

ε2, P5(v) =

v22

ε2− 1

3ε2,

P6(v) =(3|v|2 − 4)v1

ε3, P7(v) =

(3|v|2 − 4)v2

ε3,

P8(v) =9|v|4 − 15|v|2 + 2

ε4.

Note that 〈P0f〉 = ρ, 〈P1f〉 = ρu1 and 〈P2f〉 = ρu2. The second order momentsform a symmetric tensor

Θ = (Θx,Θy) =(

θ11 θ12

θ12 θ22

)=

(〈P3f〉〈P4f〉〈P4f〉〈P5f〉

),

where

Θx =(

θ11

θ12

), Θy =

(θ12

θ22

),

and for the remaining moments we set

q =(

q1

q2

)=

(〈P6f〉〈P7f〉

), s = 〈P8f〉 .

The equations of mass and momentum conservation are

(5)∂tρ + div ρu = 0,

∂tρu + divΘ +1

3ε2∇ρ = 0.

Here, the divergence is applied to the rows of Θ. The equation for Θ is

(6) ∂tΘ +2

3ε2S[ρu] +

13Q[q] = − 1

ε2τ(Θ− ρu ⊗ u),

where

S[u] =12

(2∂xu1 ∂yu1 + ∂xu2

∂yu1 + ∂xu2 2∂yu2

),

and

Q[q] =(

∂yq2 ∂yq1 + ∂xq2

∂yq1 + ∂xq2 ∂xq1

).

Finally, the third and fourth order moments satisfy

(7)∂tq +

1ε2

div(

θ22 2θ12

2θ12 θ11

)+

16∇s = − 1

ε2τq,

∂ts +4ε2

divq = − 1ε2τ

s.

Altogether, we obtain a hyperbolic system with stiff relaxation terms. The deter-mination of the diffusion limit of the above system is straightforward. From themomentum equation in (5) we conclude that ∇ρ tends to zero as ε → 0. Hence, ρ



approaches a constant ρ (which is the Boussinesq relation in the isothermal case).Writing ρ = ρ(1 + 3ε2p), equation (5) transforms into

(8)∂tp +

13ε2

divu = − div (pu),

∂tu + div1ρΘ + ∇p = −3ε2∂t(pu).

For ε → 0, equation (6) yields at the lowest order

(9)1ρΘ = u ⊗ u − 2τ

3S[u].

Since (7) decouples completely from the other equations (in lowest order) and since2 divS[u] = (∆ + ∇div )u, we obtain from (8) and (9) the incompressible Navier-Stokes equations as a limiting system

(10)divu = 0,

∂tu + divu ⊗ u + ∇p =τ

3∆u,

where the Reynolds number is related to the relaxation time by Re = 3/τ .We remark that (5), (6) and (7) can be viewed as a relaxation system for the

Navier-Stokes equations (10).

2.2. Simplified relaxation systems. We consider the system of equations in (6)and (8). For numerical reasons we simplify this system in such a way that the limitas ε tends to zero is preserved, i.e. is the same as in the original system, (6) and(8).

From equation (8) we neglect the term − div pu and −3ε2∂tpu. From equation(6) we neglect the term 1

3Q[q] and introduce a new term, ∇a[u], as follows:

For p,u = (u1, u2) and Θ = (Θx,Θy) =(

θ11 θ12

θ12 θ22

)as defined above, we

consider the system

(11)

∂tp +1ε2

divu = 0,

∂tu + divΘ +1ρ∇p = 0,

∂tΘ + ∇a[u] +2ε2

Sε[u] = − 1ε2τ

(Θ− u ⊗ u),

where

Sε[u] = S[u] − ε2

2∇a[u].

We have added and subtracted the term

∇a[u] =(a2∂xu, b2∂yu

)=

(a2∂xu1 b2∂yu1

a2∂xu2 b2∂yu2

),

where a2∂xu =(

a2∂xu1

a2∂xu2

)and b2∂yu =

(b2∂yu1

b2∂yu2

)with a =

(ab

)∈ R

2+. Obviously

the limit equations for this system are again the incompressible Navier-Stokes equa-tions with Reynolds number Re = 1/τ .



Remark 1. Considering the nonstiff advection parts in (11) separately for u andΘ we obtain a hyperbolic system with characteristic speeds ±a and ±b in x and ydirection

(12)∂tu + divΘ = 0,

∂tΘ + ∇a[u] = 0.

As we will see in section 4, a is chosen depending on the local speeds.

We can build a second relaxation system by neglecting the time derivative in thefirst equation of (11) without altering the other equations. This is practical if oneconsiders an implementation in the vorticity formulation as given below. To thisaim we consider the system

(13)

divu = 0,

∂tu + divΘ +1ρ∇p = 0,

∂tΘ + ∇a[u] = − 1ε2τ

(Θ− u ⊗ u + 2τSε[u]).

We introduce the vorticity ω = ∂xu2 − ∂yu1 by taking a two-dimensional curl ofthe second and third equation in (13) and applying the divergence-free condition,divu = 0. A relaxation system for vorticity is then derived as

∂tω + divΦ = 0,(14)

∂tΦ + ∇a[ω] = − 1ε2τ

(Φ− ωu + 2τ∇ω),

where

ω ∈ R, Φ = ∇× Θ =(

ϕ1

ϕ2

)∈ R

2, u =(

u1

u2

)∈ R

2, ∇a[ω] =(

a2∂xωb2∂yω

).

For Θ the curl is taken row-wise. Then u ∈ R2 is determined by solving the Poisson

problem

∆ψ = ω, u = ∇⊥ψ =(−∂yψ∂xψ

),

with ψ denoting the stream function. In the latter relaxation equations (13) thewhole system reduces to only three equations (for the variables ω, ϕ1, ϕ2) insteadof six (for the variables u1, u2, θ11, θ12, θ21, θ22) as in the case of primitive variables.

Having developed the Lattice-Boltzmann type relaxation systems (equation (11),(13) or (14)), what remains is to develop high-order relaxation schemes which inthe limit as ε → 0 converge uniformly to numerical schemes for incompressibleNavier-Stokes equations. We will present a full discussion on some schemes thatcan be used to compute the relaxation systems in Section 3 below.

3. Numerical schemes

To develop numerical schemes for the relaxation systems developed in Section2.2 above, we will consider equation (11). This equation is used as a case study forour derivations since to develop numerical schemes for the other systems, namely(13) and (14), one needs analogous manipulations and neglecting similar terms aspresented above.



3.1. Space discretizations. In this section high order upwind discretizations aredeveloped for the nonstiff advection part in (11). The stiff part is treated by highorder centered differences, as in [24, 25]. In the remainder of this section the timecontinuous version of the scheme is considered (method of lines). The full space-time discretization is obtained combining the spatial discretization obtained herewith the time discretization described in the next subsection.

To discretize the equations in space we use a uniform grid in the x- and y-directions with grid points (xi, yj) with spacing h. Consider the nonstiff linear partof the system in equations (11) as presented in (12). One observes that for the x-direction Θx ±au are the characteristic variables associated with the characteristicspeeds ±a. For the y-direction the characteristic variables associated with thecharacteristic speeds ±b are Θy±bu. According to these considerations the values ofthe characteristic variables are determined at cell-boundaries following the approachin [26]. This can be done in a straightforward way for a second order method. Fora third order method we use for the reconstruction step a third order CWENOinterpolant [1]. Similar reconstructions were also applied in [2, 3]. We report forthe convenience of the reader the polynomials, pij(z; x), for the reconstruction in2D in the x-direction. These polynomials, in the cell (i, j) with cell-center (xi, yj),in the case of a second order method, are given by

pij(z; x) = zij + sij(x − xi),(15)

where the MinMod limiter in sij ,

sij(z) =1h

MinMod(zij − zi−1j , zi+1j − zij),

is applied componentwise. The variable zij denotes the cell average of a vectorfunction z(x, y) in the cell (i, j) taken from a set, z = {zij}, for all cells (i, j) in thecomputational domain.

For the third order CWENO case using Simpson’s rule we have

pij(z; x) = wLPLij(z; x) + wRPR

ij(z; x) + wCPCij(z; x),(16)

with

PRij(z; x) = zij +

1h

(zi+1j − zij)(x − xi),

PLij(z; x) = zij +

1h

(zij − zi−1j)(x − xi),

and

PCij(z; x) = zij −

112

(zi+1j − 2zij + zi−1j) −112

(zij+1 − 2zij + zij−1)

+12h

(zi+1j − zi−1j)(x − xi) +1h2

(zi+1j − 2zij + zi−1j)(x − xi)2.

In expression (16) for k = L, R, C

wk =αk∑l αl

, αk =ck

(γ + ISk)β,



and

cL =14, cR =

14, cC =

12,

ISL = (vij − vi−1j)2, ISR = (vi+1j − vij)2,

ISC =133

(vi+1j − 2vij + vi−1j)2 +14(vi+1j − vi−1j)2,

with γ = 10−6, β = 2. Clearly any other high order reconstruction procedureapplies.

We proceed further with the MUSCL approach as in [26] to determine the char-acteristic variables at the boundary of the cells [xi−1/2, xi+1/2]

(Θx + au)i+1/2j = pij(Θx + au; xi+1/2),

(Θx − au)i+1/2j = pi+1j(Θx − au; xi+1/2).

An analogous procedure is used for the y direction and (Θy ± bu)ij+1/2.We denote by F(1)

h ,F(2)h the discretization of the convective parts divΘ and

∇a[u] in equation (11), respectively. They are described as follows:Using the reconstruction polynomial given above componentwise one obtains

F(1)h (Θ,u) =

1h

(Θxi+1/2j − Θx

i−1/2j) +1h

(Θyij+1/2 − Θy

ij−1/2),

and

F(2)h (Θ,u) =

(1h

(a2ui+1/2,j − a2ui−1/2j),1h

(b2uij+1/2 − b2uij−1/2))

,

where the numerical fluxes are given by

Θxi+1/2j =

12(Θx

ij + Θxi+1j) −

a

2(ui+1j − uij)

+12(σ(1)

ij (Θx + au) − σ(1)i+1j(Θ

x − au)),

Θyij+1/2 =

12(Θy

ij + Θyij+1) −

b

2(uij+1 − uij)

+12(σ(2)

ij (Θy − bu) − σ(2)ij+1(Θ

y − bu)),

ui+1/2j =12(uij + ui+1j) −

12a

(Θxi+1j − Θx

ij)

+12a

(σ(1)ij (Θx + au) + σ

(1)i+1j(Θ

x − au)),

uij+1/2 =12(uij + uij+1) −

12b

(Θyij+1 − Θy

ij)

+12b

(σ(2)ij (Θy + bu) + σ

(2)ij+1(Θ

y + bu)).

Here a and b can be chosen locally; see the last section for further details. In thesecond order case the σ

(k)ij , k = 1, 2, are given by

σ(1)ij (z) =

12MinMod(zij − zi−1j , zi+1j − zij),

and

σ(2)ij (z) =

12MinMod(zij − zij−1, zij+1 − zij).



In the third order case the CWENO reconstruction gives

σ(1)ij (z) =

wR

2(zi+1j − zij) +

wL

2(zij − zi−1j) +

wC

4(zi+1j − zi−1j)

−wC

12(zi+1j − 2zij + zi−1j) −

wC

12(zij+1 − 2zij + zij−1)

+wC

4(zij+1 − 2zij + zij−1),

and

σ(2)ij (z) =

wR

2(zij+1 − zij) +

wL

2(zij − zij−1) +

wC

4(zij+1 − zij−1)

−wC

12(zij+1 − 2zij + zij−1) −

wC

12(zi+1j − 2zij + zi−1j)

+wC

4(zi+1j − 2zij + zi−1j).

Now we must discretize the pressure variable and the stiff parts, i.e. terms with

the coefficient1ε2

, in equations (11). First, we denote the discrete gradient byGh and the discrete divergence by Dh. They are given by second or fourth ordercentered differences, respectively. Sε

h and Sh denote second or fourth order centereddifference approximations of Sε and S.

Finally, we obtain a high order spatial discretization for the moment systemcharacterized by

p +1ε2

Dh · u = 0,

u + F(1)h (Θ,u) + Ghp = 0,(17)

Θ + F(2)h (Θ,u) +

2ε2

Sεh(u) = − 1

ε2τ(Θ− u ⊗ u),

or equivalently

Dh · Ghp − 2ε2p = −Dh · F(1)h (Θ,u),

u + F(1)h (Θ,u) + Ghp = 0,

Θ + F(2)h (Θ,u) = − 1

ε2τ(Θ− u ⊗ u + 2τSε

h(u)).

A corresponding high order upwind based space discretization for the incompress-ible Navier-Stokes equations is obtained by considering the limit of the above dis-cretization as ε → 0:

Dh · Ghp = −Dh · F(1)h (u),

u = −F(1)h (u⊗ u − 2τSh(u),u) − Ghp.

3.2. Time discretizations. To treat only the limit equations (ε = 0) we could useany explicit high order Runge-Kutta method combined with a Poisson solver andthe limiting (relaxed) spatial discretization. The Poisson equation is in this caseonly used to determine the divergence-free velocities via ∇p and not to advancethe pressure for one time step. For example, the maximum preserving schemes in[34, 43] can be used or the DUMKA scheme [36] or simply a suitable Runge Kuttamethod. Several numerical tests in this direction are presented in the last section.

We need to point out that it is necessary to be careful if an explicit time dis-cretization is to be used. Due to the diffusion terms, standard explicit schemes



would suffer from excessive time step restrictions since in that case ν∆t ≤ ch2 fora suitable constant c of order unity, which depends on the scheme. This is whereDUMKA, as an explicit scheme, is advantageous to use since it has a large value ofc; hence larger time steps are admissible.

Further, a more challenging issue is to obtain a discretization of the relaxationsystem that works uniformly for all ranges of the parameter ε, thus allowing us tostudy the numerical passage from the discrete velocity models for the Boltzmannequation to the INS regime.

To this aim we can use implicit-explicit (IMEX) Runge-Kutta methods of thetype developed in [4, 38]. In these schemes the nonstiff parts are treated explicitlyand the stiff ones are treated implicitly. In particular, we will treat the pressure-velocity coupling in an implicit way. As we will see, in the small mean free pathlimit ε → 0 this leads to a projection scheme for the incompressible Navier Stokesequations. First, second and third order time discretizations are discussed in thesequel.

We denote the time step by k and use the superscript n to denote the timeiterations. For the first order method we can use the following simple time dis-cretization:

un+1 = un − k( divΘn + ∇pn+1),

Θn+1 = Θn − k∇a[un] − k

ε2τ

(Θn+1 + 2τSε[un+1] − un+1 ⊗ un+1

),(18)

pn+1 = pn − k

ε2divun+1.

Using the last equation in (18) into the first equation in (18) yields a Helmholtzequation for the pressure

∆pn+1 − ε2

k2pn+1 =

1k

divun − div div Θn − ε2

k2pn.(19)

This equation can be solved by a suitable iterative method. Then un+1 is deter-mined using the first equation in (18).

Obviously, as ε → 0 the time integration scheme tends to a time discretizationof the incompressible Navier Stokes equations. We obtain for ε → 0 the Poissonequation for the pressure

∆pn+1 =1k

divun − div divΘn,

together with

un+1 = un − k(divΘn + ∇pn+1),

Θn+1 = 2τS[un+1] − un+1 ⊗ un+1.

Thus, in the limit, we have obtained the usual projection method for the incom-pressible Navier-Stokes equations; see, e.g. [39]. We note that the incompressibilitycondition is fulfilled for u in every time step.

In IMEX notation the above first order scheme is given by the explicit andimplicit Butcher tableau [4, 38]

0 0 01 1 0

1 0

0 0 01 0 1

0 1



For the above semi-implicit time discretization the usual hyperbolic and parabolicCFL conditions have to be fulfilled to guarantee stability.

For the second order time discretization we choose a two stage IMEX RungeKutta method [38, 4] which guarantees second order accuracy in the stiff limit.The associated explicit and implicit Butcher tableau are

0 0 0 0γ γ 0 01 δ 1 − δ 0

δ 1 − δ 0

0 0 0 0γ 0 γ 01 0 1 − γ γ

0 1 − γ γ

with γ = 1 −√

2/2 and δ = 1 − 1/2γ. This yields

Step 1:

∆pn+1/2 − ε2

k2γ2pn+1/2 = − ε2

k2γ2pn +

1kγ

divun − div divΘn,

un+1/2 = un − kγ(divΘn + ∇pn+1/2

),

Θn+1/2 = Θn − kγ∇a[un]

− kγ

ε2τ

(Θn+1/2 − un+1/2 ⊗ un+1/2 + 2τSε[un+1/2]

).

Step 2:

∆pn+1 − ε2

k2γ2pn+1 = − ε2

k2γ2pn +

1kγ

((1 − γ)divun+1/2 + γdivun

)−

(δdiv divΘn + (1 − δ)div divΘn+1/2

)− 1 − γ

γ∆pn+1/2,

un+1 = un − k(δdivΘn + (1 − δ)divΘn+1/2

)− k

((1 − γ)∇pn+1/2 + γ∇pn+1

),

Θn+1 = Θn − k(δ∇a[un] + (1 − δ)∇a[un+1/2]

)

− k

ε2τ

((1 − γ)

(Θn+1/2 − un+1/2 ⊗ un+1/2 + 2τSε[un+1/2]

)

+ γ(Θn+1 − un+1 ⊗ un+1 + 2τSε[un+1]

)).

For ε → 0 we obtain a second order time discretization of the INS equations basedon the explicit scheme in the above IMEX method. Note that the divergence freecondition is guaranteed in the limit in every time step.

A third order method is developed based on a third order IMEX scheme [4, 38].The associated explicit and implicit tables are



0 0 0 0 0 012

12 0 0 0 0

23

1118

118 0 0 0

12

56 −5

612 0 0

1 14

74

34 −7

4 014

74

34 −7

4 0

0 0 0 0 0 012 0 1

2 0 0 023 0 1

612 0 0

12 0 −1

212

12 0

1 0 32 −3

212

12

0 32 −3

212

12

The scheme was selected based on the fact that it seemed to have reasonable con-vergence and stability properties based on the results presented in [38]. We omitfor brevity the details of the scheme.

Remark 2. In the previous schemes it is of paramount importance that the implicitpart of the time integrator is diagonally implicit. Fully implicit schemes (for whichthe Butcher tableau of the implicit part contains nonzero elements above the maindiagonal) originate systems of nonlinear algebraic equations that need to be solvedusing suitable iterative techniques.

Remark 3. In order to develop a relaxed scheme for the limit INS equations only,one may use the simplified relaxation system (13).

4. Numerical results and examples

In this section we test the above schemes in several different situations. Our testexamples can be considered in four general categories: starting with the1-D time dependent problem we first perform an accuracy test for the third or-der scheme; second, we test if the scheme converges uniformly to the limit ε → 0; inpart 2 we consider a stationary problem in 2-D and test the ability of our scheme toresolve discontinuous, solutions especially for the convection terms; and in part 3the scheme is tested on time dependent incompressible Navier-Stokes problems in2-D. Indeed with such a simplified approach, it will be demonstrated that competi-tive schemes for solving incompressible flow problems have been developed.

4.1. Accuracy and convergence for 1-D test problems. Accuracy and con-vergence of the scheme developed in the previous section is numerically investigatedfor the relaxation system leading to the inviscid and viscous Burgers equation inone space dimension, respectively. That means we consider a system analogous to(13):

∂tu + ∂xΘ = 0,(20)

∂tΘ + a∂xu = − 1ε2

(Θ − 12u2 + τ∂xu).

We consider the third order method developed above based on the CWENO recon-struction and the third order IMEX scheme.

Test 1: AccuracyIn order to check the accuracy of our third-order relaxation scheme, we first

consider the one-dimensional inviscid Burgers equation (τ = 0 and ε = 0). We solvethe equations (20) in [0, 2π] augmented with the smooth initial data, u(x, 0) = 0.5+sin(x), and periodic boundary conditions. We discretize the spatial domain into Ngridpoints, and we choose the relaxation parameter a = 1.5 in all computations.Recall that the unique entropy solution of (20) is smooth up to the critical time



T = 1. In Table 1 we show the error norms at the pre-shock time t = 0.5 when thesolution is still smooth using CFL = 0.75. The errors are measured by the differencebetween the pointvalues of the exact solution and the reconstructed pointvaluesof the computed solution. As expected our scheme preserves the third order ofaccuracy.

Table 1. Error-norms for the invscid Burgers problem.

N L∞-error Rate L1-error Rate L2-error Rate

40 0.37681E-01 —– 0.28977E-01 —– 0.30533E-01 —–80 0.15964E-01 1.239 0.71792E-02 2.013 0.82323E-02 1.891160 0.47363E-02 1.753 0.12559E-02 2.515 0.17511E-02 2.233320 0.78772E-03 2.588 0.14477E-03 3.117 0.22551E-03 2.957640 0.69819E-04 3.496 0.92831E-05 3.963 0.17196E-04 3.7131280 0.65638E-05 3.411 0.61968E-06 3.905 0.13613E-05 3.659

Test 2: Uniform ConvergenceIn this example we investigate the uniform convergence behavior of the relax-

ing method for different values of ε. The uniform convergence of the method isnumerically investigated for the relaxation system, equation (20), leading to theone-dimensional viscous Burgers equation. Once again we consider the third ordermethod developed above based on the CWENO reconstruction and the third orderIMEX scheme.

We plot the convergence rates for different values of ε in Figure 1. The ratesare determined by comparing the errors at time T = 1 computed from ∆t = 0.01,∆t = 0.005, ∆t = 0.0025, and ∆t = 0.00125, respectively.

In space the energy norm is used. Third order accuracy is reached for very smalland very large values of ε, whereas for intermediate values a slight deterioration ofthe accuracy is observed. This is expected and in good agreement with the resultsobtained in [38].

10 10 5 10 4 10 3 10 2 10 1 1000.5

1

1.5

2

2.5

3

3.5

4

4.5

ε

Con

verg

ence

Rat

e

ν = 0.00001

u Component θ Component

10 10 5 10 4 10 3 10 2 10 1 1000.5

1

1.5

2

2.5

3

3.5

4

4.5

5

ε

Con

verg

ence

Rat

e

ν = 0.001

u Component θ Component

Figure 1. Convergence rates for τ = 0.00001 and τ = 0.001



Figure 2. The setup of the step profile. The z-axis depicts ‖u‖.

4.2. Stationary 2-D test problems. The first problem is a simple test case tocompare the spatial discretization of the convective term with the discretizationgiven by other methods. We compare the second and third order relaxed schemeswith different central methods [34, 31]. The problem we consider is the stationaryconvection-diffusion problem [16]

div (u⊗ u) = τ∆u

on [0, 1]2. Here we consider the relaxed schemes ε → 0. For these stationaryproblems our main focus will be on the qualitative behaviour of our schemes toresolve the solutions with very sharp gradients.

The first example is pure convection of a step profile [16]. More details on howthis system is discretized and treated numerically can be found in [16]. This is asimple but good test problem for examining the relative performance of differentnumerical approximations to convection terms with sharp gradients for u1 and u2.We consider (x, y) in [0, 1]2. The computation domain is divided into two sub-domains which give a step profile as sketched in Figure 2.

The flow profile inside the domains is extended to the boundary. The computa-tional domain is discretized using a 41 × 41 regular mesh for different flow anglesθ. Values for τ were also varied. The resulting nonlinear system is solved by theNewton method using a GMRES-based solver described and implemented in [27].

The results are plotted in Figure 3. They show the computed profile at the linex = 1

2 for both the velocity components u1 and u2. In the figures we make a com-parison of the following second and third order schemes: Kurganov and Tadmor’s(KT) second order scheme [34]; Kurganov and Levy’s (KL) third order scheme [1];the relaxed scheme based on Jin and Xin’s (JX) second order approach [26] andthe new third order relaxed scheme (RKL) based on CWENO reconstruction.

In this and the following tests we have used the local characteristic speeds as in[34, 1] to define a at the point (xi, yj):

ai+ 12 j = 2 max{|pij(u1; xi+ 1

2)|, |pi+1j(u1; xi+ 1

2)|},

bij+ 12

= 2 max{|pij(u2; yj+ 12)|, |pij+1(u2; yj+ 1

2)|},



0 0.2 0.4 0.6 0.8 10.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

y

u 1

Pure convection of a Step Profile, θ = 45°, τ = 0

exactKTJXKLRKL

0.3 0.4 0.50.7

0.75

0.8

0.85

Zooming

0 0.2 0.4 0.6 0.8 10.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

y

u 1

Pure convection of a Step Profile, θ = 25°, τ = 0

exactKTJXKLRKL

0.5 0.551.5

1.6

1.7

1.8

Zooming

Figure 3. Comparison of different approximations of a stationarystep profile for u1 with different angles θ = π/4, 25 degrees on theleft and right, respectively.

|U| = 1

(0,1) (1,1)

(0,0) (1,0)X

Y45

y*y*

y*y*

v

u

o

|U| = 2

|U| = 1

00.2

0.40.6

0.81

00.2

0.40.6

0.811

1.2

1.4

1.6

1.8

2

x

The Box Profile

y

Figure 4. The setup of the box profile. The z-axis depicts ‖u‖.

0 0.2 0.4 0.6 0.8 10.7

0.8

0.9

1

1.1

1.2

1.3

1.4

y

u 2

Pure convection of a Box Profile, θ = 45°, τ = 0, y* = 0.15

exactKTJXKLRKL

0.4 0.5 0.61.34

1.36

1.38

Zooming

0 0.2 0.4 0.6 0.8 10.7

0.8

0.9

1

1.1

1.2

1.3

1.4

y

u 2

Pure convection of a Box Profile, θ = 45°, τ = 0, y* = 0.2

exactKTJXKLRKL

0.4 0.61.36

1.38

1.4

Zooming

Figure 5. Comparison of different approximations for convectionin stationary flow with a box profile.



and ai− 12 j , bij− 1

2in the same way. Here the polynomials pij are defined for scalar

variables u1 and u2 analogous to the definition in (15) or (16). The results for thestep profile in Figure 2 are shown in Figure 3.

In this figure profiles for the u1 are shown for τ = 0 and are compared with theexact stationary profile [16]. The profiles for u2 are similar. Tests on small valuesof τ = 10−6 were made, and the results are also similar. The second exampleis pure convection of a box profile. We consider a box-shaped profile as shownin Figure 4. This example was also presented in [16]. It is normally selectedbecause the severe, rapid change in the gradients in velocity resembles, in differentways, many similar profiles found, for example, in practical flows in which severepeak profiles are characteristic across shear layers. As in the previous example wecompare approximations of the profile across a vertical plane in the middle of thesolution domain. We compare the different schemes using a uniform 41× 41 mesh,and the results are presented in Figure 5.

In Figure 5, we present results for stationary profiles of u2. The results for u1

are very similar.Here too, since we are mainly investigating the performance of the approximation

on the convection terms, results for τ = 0 are presented. As one observes from thefigures the second and third order relaxed methods give results that are qualitativelysimilar to the KT and KL scheme, respectively. In some cases, the relaxed schemeis more accurate, and in other cases RKL is more accurate.

4.3. Instationary 2-D incompressible Navier-Stokes test problems. Herewe are especially interested in the results of the relaxing method for very small ε andthe results of the relaxed method setting ε = 0 for the incompressible Navier-Stokesequations.

Problem 1: (Shear layer). The next problem is set up to test the behavior ofthe discretization for nonstationary situations when steep gradients are involved.We consider the following periodic problem [1]. Let (x, y) ∈ [0, 2π]2. The initialconditions are

u(x, y, 0) =

⎧⎨⎩

tanh( 1ρ (y − π/2)), y ≤ π,

tanh( 1ρ (3π/2 − y)), y > π,

and

v(x, y, 0) = δ sin(x),

with δ = 0.05 and ρ = π/15. We use 64 × 64 and 128 × 128 spatial grid points,respectively. The vorticity system (14) and the associated relaxing method is used.

First we consider the situation in the incompressible Navier Stokes/Euler limit.We compute the solution using the relaxing scheme with ε = 10−6 combined withan IMEX method of first, second and third order.



Figure 6 shows the evolution at time t = 4 for the first, second and third ordermethods described in the paper for the Euler case (τ = 0 in (14)) and ε = 10−6.Figure 7 shows the evolution at time t = 10 for the first, second and third ordermethods for the Navier-Stokes case with τ = 0.01 and ε = 10−6.

To compare these results qualitatively with other methods, see for example [1].A closer convergence study for the vorticity variable yields the results displayedin Tables 2 and 3. The results show the second and third order of the schemes,respectively, in the shear layer case. In the second order case we have used a van Leerlimiter instead of the minmod limiter. Minmod gives slightly worse results. As areference solution we used the solution obtained on the finest mesh of 528 × 528gridpoints.

Finally, the situation with large ε is considered. Figure 8 shows the evolution attime t = 10 for the third order methods for the Navier-Stokes case with τ = 0.01and ε = 0.1 and for comparison ε = 10−6.

Table 2. Error-norms for the double shear layer problem withτ = 0 and ε = 10−6 at t = 2.

Gridpoints L∞-error Rate L1-error Rate L2-error Rate

16 × 16 3.15715E-01 —– 5.79035E-01 —– 4.26853E-01 —–32 × 32 8.54186E-02 1.886 1.79334E-01 1.691 1.26904E-01 1.75064 × 64 2.23542E-02 1.934 4.91291E-02 1.868 3.39324E-02 1.903

128 × 128 5.58856E-03 2.000 1.26978E-02 1.952 8.50078E-03 1.997264 × 264 1.29906E-03 2.103 3.10696E-03 2.031 1.98562E-03 2.098

Table 3. Error-norms for the double shear layer problem withτ = 0 and ε = 10−6 at t = 2.

Gridpoints L∞-error Rate L1-error Rate L2-error Rate

16 × 16 1.12035E-02 —– 1.87151E-02 —– 1.62533E-02 —–32 × 32 1.62774E-03 2.783 2.88411E-03 2.698 2.45659E-03 2.72664 × 64 2.16565E-04 2.910 4.00016E-04 2.850 3.35331E-04 2.873

128 × 128 2.74676E-05 2.979 5.23789E-05 2.933 4.33645E-05 2.951264 × 264 3.42870E-06 3.002 6.54736E-06 3.000 5.42057E-06 3.000



0 1 2 3 4 5 60

1

2

3

4

5

6

h = 1/64

0 1 2 3 4 5 60

1

2

3

4

5

6

h = 1/128

0 1 2 3 4 5 60

1

2

3

4

5

6

h = 1/64

0 1 2 3 4 5 60

1

2

3

4

5

6

h = 1/128

0 1 2 3 4 5 60

1

2

3

4

5

6

h = 1/64

0 1 2 3 4 5 60

1

2

3

4

5

6

h = 1/128

Figure 6. Results for the Euler case: first order (top), secondorder (medium) and third order (bottom).



0 1 2 3 4 5 60

1

2

3

4

5

6

h = 1/64

0 1 2 3 4 5 60

1

2

3

4

5

6

h = 1/128

0 1 2 3 4 5 60

1

2

3

4

5

6

h = 1/64

0 1 2 3 4 5 60

1

2

3

4

5

6

h = 1/128

0 1 2 3 4 5 60

1

2

3

4

5

6

h = 1/64

0 1 2 3 4 5 60

1

2

3

4

5

6

h = 1/128

Figure 7. Results for the Navier-Stokes case: first order (top),second order (medium) and third order (bottom).



0 1 2 3 4 5 60

1

2

3

4

5

6

0 1 2 3 4 5 60

1

2

3

4

5

6

Figure 8. Results for the Navier-Stokes case: third order, ε = 0.1and ε = 10−6.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Re = 1000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Re = 10000

Figure 9. Driven Cavity, plot of the stream function for Re =1000 and Re = 10000.

Problem 2: (Driven cavity). Finally, we consider a driven cavity situation withx ∈ [0, 1]2 and the usual boundary conditions with a drift u = (u, 0) parallel to theboundary at the top of the square and u = 0 at the other sides. For the simulationwe use the third order relaxation method developed above based on the relaxationsystem (14). Zero initial conditions are used for all moments. u is chosen equalto 1. ε is again chosen equal to 10−6 for the relaxing scheme. We use 128 × 128spatial grid points. The time step is chosen according to the CFL condition. Figure9 shows a plot of the stream-functions for Re = 1000 at t = 100 and Re = 10000at t = 1000.

5. Conclusions

1. Several relaxation systems based on a Lattice Boltzmann type discretevelocity model have been presented. In the diffusive limit the systemsrelax towards the incompressible Navier-Stokes equations.

2. Second and third order relaxation schemes working uniformly in the incom-pressible Navier-Stokes limit have been presented.



3. The space discretization is obtained using second and third order upwinddiscretization based on slope limiters and CWENO discretizations.

4. For the time discretization high order IMEX Runge Kutta methods havebeen used to obtain uniform accuracy with respect to the stiff relaxationtime [38].

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School of Mathematical Sciences, University of KwaZulu-Natal, Private X01, 3209

Pietermaritzburg, South Africa

E-mail address: [email protected]

Fachbereich Mathematik, TU Kaiserslautern, Erwin-Schroedinger-Str. 48, D-67663

Kaiserslautern, Germany


Department of Mathematics, University of Ferrara, Via Machiavelli 35, I-44100 Fer-

rara, Italy


Fachbereich Mathematik, TU Kaiserslautern, Erwin-Schroedinger-Str. 48, D-67663

Kaiserslautern, Germany



Lattice-Boltzmann type relaxation systems and high order relaxation schemes for the incompressible Navier-Stokes equations

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